Testing rebalancing strategies for
stock-bond portfolios across
different asset allocations
Hubert Dichtl  Wolfgang Drobetz  Martin Wambach
HFRC Working Paper Series No.4 – 5. August 2014
Hamburg Financial Research Center e.V.  c/o Universität Hamburg | Von-Melle-Park 5  20146 Hamburg
Tel. 0049 (0)40 42838 2421 | Fax: 0049 (0) 42838 4627 | info@hhfrc.de  www.hhfrc.de
Testing Rebalancing Strategies for Stock-Bond Portfolios
Across Different Asset Allocations
Hubert Dichtl, Wolfgang Drobetz, and Martin Wambach
We compare the risk-adjusted performance of stock-bond portfolios between rebalancing and buy-and-hold across different asset allocations by reporting statistical significance levels. Our investigation is based on a 30-year dataset and incorporates the financial markets of the United States, the United Kingdom, and
Germany. To draw reasonable recommendations to investment management, we
implement a history-based simulation approach which enables us to mimic realistic market conditions. If the portfolio weight of stocks exceeds 30%, our empirical results show that a frequent rebalancing significantly enhances risk-adjusted
portfolio performance for all analyzed countries and all risk-adjusted performance measures.
Keywords: rebalancing, stock-bond portfolio, bootstrap, statistical inference
Hubert Dichtl is managing director at alpha portfolio advisors GmbH in Bad Soden/Ts, Germany. Wolfgang Drobetz is a professor of the Department of Business at the University of
Hamburg. Martin Wambach is a research fellow of the Department of Business at the University of Hamburg.
Corresponding author: Wolfgang Drobetz (wolfgang.drobetz@wiso.uni.hamburg.de)
2
1.
Introduction
Strategic asset allocation constitutes an integral component of portfolio performance. Analyzing the impact of strategic asset allocation on portfolio performance without removing the
effects of market movements, Brinson et al. (1986), Brinson et al. (1991), as well as Ibbotson
and Kaplan (2000) all provide evidence that 100% of the return level and about 90% of the
variation of a fund’s returns over time is explained by its investment policy. While the first
result states an identity in aggregate market conditions derived from the fact that active management must be a zero sum game by definition, the second result does not differentiate between the effect of overall market movements and the impact of strategic asset allocation on
the variation of a fund’s return. Accounting for the impact of market movements, Xiong et al.
(2010) document that strategic asset allocation and active management are equally important
for explaining return differences on aggregate. Although strategic asset allocation is less important than predicted by prior studies, the findings of Xiong et al. (2010) reconfirm the rationale that the investment policy is of considerable importance to investment practice.
Therefore, building on the fundamentals of modern portfolio theory by Markowitz (1952),
many optimization routines have been developed by both academic researchers and practitioners. The main objective is the construction of strategic portfolios that meet the specific
requirements of different investors. Developments based on the mean-variance portfolio optimization of Markowitz (1952) are still the best known and most widely used portfolio optimization approaches in practice. However, the implementation of mean-variance optimizers
raises several problems. Michaud (1989) and Kritzman (2006) report that mean-variance optimization routines tend to under- and overweight assets or even entire asset classes with the
highest input errors, often resulting in a non-optimal risk-adjusted performance. Moreover, if
assets are close substitutes for one another, small errors of the input parameters could result
in significant misallocations of the portfolio. For all these reasons, other more robust optimization routines have been developed (Black and Litterman, 1992; Fabozzi et al., 2007; DeMiguel et al., 2009a, 2009b). Instead of focusing on portfolio optimization techniques, this paper
analyzes the risk-adjusted performance of stock-bond portfolios by systematically changing
the underlying portfolio weights. Given an initial asset allocation, we additionally investigate
whether a frequent rebalancing contributes to enhance risk-adjusted portfolio performance.
Although this question is of utmost importance for investment practice, prior research on
portfolio rebalancing has lacked a focus on investigating different asset allocations. For ex-
3
ample, while Arnott and Lovell (1993) consider a 50/50 asset allocation, Harjoto and Jones
(2006), Tokat and Wicas (2007) and Jaconetti et al. (2010) all concentrate on a 60/40 stockbond portfolio. Only Tsai (2001) examines the impact of rebalancing on the risk-adjusted
performance across different asset allocations. In particular, she analyses the Sharpe ratio of
five different stock-bond portfolios over the sample period from January 1986 to December
2000. Considering seven asset classes, the equity portfolio weight varies between 20%, 40%,
60%, 80%, and 98%, respectively. While it is not obvious which specific rebalancing algorithm should be adopted, all four rebalancing strategies under investigation generate a higher
Sharpe ratio for all five portfolios in comparison to buy-and-hold. However, the explanatory
power of her analysis is weakened by the fact that it is based on one single 15-year period,
which intensifies the potential problem of data snooping. It could be the case that the empirical results are substantially driven by specific characteristics of the underlying sample period
while the investment strategy under investigation could have only a minor impact on the riskadjusted portfolio performance (Brock et al., 1992).
In order to shed light on this issue, we compare the risk-adjusted performance of a given
asset allocation between buy-and-hold and rebalancing strategies on a statistical basis. In particular, we use the stationary bootstrap of Politis and Romano (1994). Preserving most of the
time series characteristics, this history-based simulation approach enables us to report statistical significance levels under realistic market conditions.
Our results provide evidence that rebalancing significantly outperforms buy-and-hold if
the portfolio weight of stocks exceeds a certain threshold. Depending on the country under
investigation and on the risk-adjusted performance measure, this threshold ranges between
0% and 30%. We further document that the optimized asset allocation is subject to both the
country and the period under investigation.
2.
Implemented Rebalancing Strategies
As shown in Table 2, we focus on ten different rebalancing strategies, which are categorized
by the underlying rebalancing algorithm into four distinctive rebalancing classes: (i) buy-andhold, (ii) periodic rebalancing, (iii) threshold rebalancing, and (iv) range rebalancing. If no
rebalancing period is defined, rebalancing reduces to buy-and-hold, i.e., no transactions take
place during the entire investment period until divestment. In contrast, periodic rebalancing is
characterized by a reallocation to the initial portfolio weights at the end of each predetermined period. We consider yearly, quarterly, and monthly trading frequencies. Transaction
4
costs are quoted at 15 bps per roundtrip throughout the entire analysis. They can be composed
in 10 bps for buying/selling stocks and 5 bps for buying/selling government bonds.
To reduce portfolio turnover and thus save transaction costs, a no-trade region around the
target weights can be implemented as well. In line with Norway’s Government Pension Fund
Global, this study uses a 3% symmetric no-trade interval around the target weights (Norwegian Ministry of Finance, 2012). Given the strategic decision to conduct interval rebalancing,
two different approaches must be distinguished. While threshold rebalancing requires a readjustment to the target weights at the end of each predetermined period, range rebalancing
further reduces portfolio turnover by rebalancing the assets back only to the nearest edge of
the no-trade interval at the end of each pre-assigned period (Leland, 1999). However, the
increased utility of saved transaction costs must be weighted against the modified risk-return
characteristics to draw valuable recommendations to portfolio management. Our historybased simulation takes both aspects into account.
[Insert Table 2 here]
The exact rebalancing procedure is explained with the help of an example: Consider a
portfolio consisting of 60% stocks and 40% government bonds with an underlying yearly
trading frequency. A ‘3% yearly threshold rebalancing strategy’ requires rebalancing the assets back to the initial 60/40 asset allocation whenever the stocks’ portfolio weight exceeds
the no-trade interval [57%, 63%] at the end of each year. In contrast, a ‘3% yearly range rebalancing strategy’ postulates a readjustment at the end of the year to 57% if the relative proportion of stocks has fallen below 57%, or to 63% if the stocks’ portfolio weight exceeds
63%. Otherwise, the portfolio weight of stocks lies within the no-trade region, implying that
no transactions are required.
3.
Data and Descriptive Statistics
To test whether the empirical findings are robust across countries, our analysis not only
comprises the financial market of the United States, but also those of the United Kingdom
and of Germany. Due to data availability, our monthly data covers the sample period from
January 1982 to December 2011. We obtain well-diversified stock and government bond total
return indices as well as Treasury bills (United States), LIBOR (United Kingdom), and
FIBOR (Germany) from Thomson Reuters Datastream. As the Treasury bills, the LIBOR,
and the FIBOR represent liquid instruments featuring a high trading volume on the secondary
5
market, they serve as proxies for the risk-free rates which are necessary to calculate the
Sharpe ratio. They all exhibit a maturity of three months. As all time series are quoted in local currency, investors of the corresponding country are not exposed to foreign exchange risk.
[Insert Table 1 here]
The descriptive statistics in Table 1 show substantial cross-country differences, thereby
legitimating the expansion of our analysis to the financial markets of the United Kingdom
and of Germany. For example, on the one hand, the German stock market exhibits the lowest
average annualized return while simultaneously featuring the highest annualized volatility of
all three countries. As a result, both the stock market of the United States as well as that of
the United Kingdom offer a better risk-return ratio in comparison to Germany during the underlying 30-year sample period. However, on the other hand, the German government bond
market exhibits the lowest volatility of all three countries under investigation. Therefore, we
expect that the optimized asset allocation of a German stock-bond portfolio exhibits a higher
proportion of government bonds in comparison to optimal stock-bond portfolios of the United
States and the United Kingdom. Furthermore, the low correlation between stocks and government bonds provides a first hint that a pronounced diversification effect will be found.
Overall, these cross-country differences stress the need to analyse the issue whether rebalancing leads to a significantly better risk-adjusted performance compared to buy-and-hold across
all three countries under investigation or not.
4.
Simulation Set-Up
The primary objective of this study is the statistical comparison of the impact between re-
balancing and buy-and-hold on the risk-adjusted performance of stock-bond portfolios across
different asset allocations.
Resampling Procedure In order to draw valuable recommendations to investment practice, we implement a history-based simulation approach. As our analysis is based on historical data, we use the stationary bootstrap of Politis and Romano (1994), which is applicable to
stationary, weakly dependent data. A distinctive feature of this block-bootstrap constitutes the
selection of blocks with different lengths, which allows preserving most of the original time
series properties. The approach is also in line with Cogneau and Zakamouline (2013), who
report that standard bootstrap techniques cannot be employed if time series returns are serially dependent. In particular, we simulate 10-year return paths of the stock, government bond,
6
and money markets for each country under investigation by randomly drawing paired blocks
of different lengths with replacement from the entire 30-year sample period and putting together these randomly drawn blocks.
The only parameter of this resampling procedure left to be determined is the probability P,
which constitutes the relevant factor of the different block lengths to be drawn. The higher P
is, the shorter the expected block length will be. However, instead of assigning an appropriate
probability to P, we exploit the reciprocal relation between the average block length and the
probability P resulting from the geometric distribution. In particular, the expected value of a
geometrically distributed random variable is its reciprocal value 1/P. In order to determine an
appropriate average block length, we apply the automatic block-length selection for the dependent bootstrap of Politis and White (2004) as well as the corrections made by Patton et al.
(2009). Overall, an average block length of 2 is recommended to both the stock and the government bond market for all three countries under investigation.1 Although Ledoit and Wolf
(2008) show that Politis and Romano’s (1994) stationary bootstrap is quite insensitive to the
choice of the average block size, we conduct an additional robustness check by using an average block length of 10 (results not reported). Our empirical findings do not change in quality.2
The 120 monthly spots of the simulated 10-year return path are consecutively filled according to the described selection process. This procedure enables us to simulate 10-year return paths that could have been realized in the past becasue all resampled paths are based on
historical return data. Not only can most of original time series characteristics be preserved,
but also the correlation across the three different asset classes, which helps us to make our
results comparable across countries.
Construction of Confidence Intervals We construct confidence intervals that are easy to
understand and straightforward to implement by following Efron and Tibshirani (1998), who
lay the foundation of our test whether rebalancing leads – on average – to a higher risk-
1
2
The implementation of the stationary bootstrap exactly follows the algorithm as decribed in Politis and
Romano (1994). This procedure also implies that the data, once the last observation is reached, wraps
around to a circle to the starting observation in order to avoid a trapezoidal weithing of the observations at
the beginning and at the end of the sample period. For example, consider a resampled block with four consecutive return observations starting at 259. Hence, this block consists of the return observations 259, 360,
1, and 2. If we apply a fixed block length of 1 and thus completely destroy the time series structure, however, statistical significance disappears, thereby demonstrating the need to preserve time series characteristics to the
greatest possible extent to be able to draw valuable recommendations to portfolio management.
7
adjusted performance compared to buy-and-hold. In particular, we test whether the mean of a
difference time series is equal to zero:
:∆
(1)
0
versus
:∆
0,
where PM denotes the risk-adjusted performance measure. In this study, we focus on the
Sharpe ratio, the Sortino ratio, and the Omega measure as appropriate risk-adjusted performance measures in order to stay focused on our primary contribution.3 Furthermore, the rebalancing strategies as well as the country under investigation need to be specified in order to
calculate the resulting difference between the risk-adjusted performances of both strategies:
(2)
∆
,
where A and B constitute rebalancing strategies as shown in Table 2. The arithmetic mean is
an appropriate point estimator of (2):
(3)
∆
.
An example best illustrates the exact computation of the confidence intervals. As described below, we compare the average risk-adjusted performance between quarterly periodic
rebalancing and buy-and-hold for the financial market of the United States by employing the
Sharpe ratio. In a first step, we apply both strategies to the simulated 10-year return path and
calculate the two corresponding Sharpe ratios. In order to test whether quarterly periodic rebalancing outperforms buy-and-hold on average, we repeat this procedure 100 times in a second step and calculate the average Sharpe ratio for both quarterly periodic rebalancing and
buy-and-hold. Accordingly, we receive one average Sharpe ratio for each strategy. In a third
step, we calculate the difference between these two averages by subtracting the average
Sharpe ratio of the buy-and-hold strategy from the average Sharpe ratio of the quarterly periodic rebalancing strategy. However, if this difference is positive, we still do not know whether quarterly periodic rebalancing outperforms buy-and-hold on average. There also is the
chance that the true value is negative. Therefore, we derive a distribution of equation (3) by
repeating the steps one to three 1,000 times:
3
Academic research as well as investment practice analyze a variety of risk-adjusted performance measures.
Nevertheless, Eling (2008) show that the selection of the performance measure is not critical to performance evaluation. Adcock et al. (2012) further provides evidence that even if returns are not normally distributed, the rank correlation of the Sharpe ratio and other performance measures is one or very close to it.
8
∆∗
(4)
∆∗
....
∆∗
∆∗
,
,
where equation (4) states the ordered difference series of the performance measure of interest.
In a last step, we construct percentile intervals as described by Efron and Tibshirani [1998]:4
∆∗
(5)
∙ ,
, ∆∗
∙ ,
.
The nominal level of α to be considered is 0.10. The null hypothesis
level if 0 ∉
is rejected at the 10%
:
∆∗
(6)
, ∆∗
.
Our empirical analysis is based on 100,000 different 10-year return paths of stocks, government bonds, and risk-free rates. Although these 10-year return paths are simulated, we use
the same paths for the entire investigation to be able to compare our simulation results across
different performance measures, countries, and changing asset allocations.
Overall, our history-based simulation approach enables us to preserve most of the time series properties as well as to conduct a systematic comparison by reporting statistical significance levels. Repeated simulations reveal that our results are stable in capturing the underlying sample patterns.
5.
Empirical Results
Although several studies do report that rebalancing provides a value-added for institutional
investors (Dichtl et al., 2013), three questions remain unanswered. First of all, there are no
studies on rebalancing with a focus on institutional investors outside the United States. However, according to Table 1, there are substantial differences in time series characteristics between the United States, the United Kingdom, and Germany. It is not clear ex ante whether
these cross-country differences have a different impact on the risk-adjusted performance of
rebalancing.
4
Our analysis focuses on a straightforward implementation of confidence intervals. As our sample period
covers 360 monthly return observations and thus can be considered a rather large sample, the construction
of percentile intervals according to Efron and Tibshirani (1998) is legitimated. Romano and Wolf (2006)
provide evidence that the studentized block bootstrap leads to an improved coverage accuracy for small
sample sizes, whereas Ledoit and Wolf (2008, 2011) suggest a studentized time series bootstrap in case of
small to moderate sample sizes.
9
Secondly, little research has been done on rebalancing across different asset allocations.
While Arnott and Lovell (1993) examine a 50/50 stock-bond portfolio, the analyses of Harjoto and Jones (2006), Tokat and Wicas (2007), and of Jaconetti et al. (2010) all focus on a
60/40 stock-bond portfolio. Only Tsai (2001) investigates the risk-adjusted performance of
rebalancing across different asset allocations. However, her analysis only provides a snapshot
of the particular state of the market because it is merely based on a single 15-year sample
period.
Thirdly, most studies presented above are based on historical analyses. However, as rebalancing constitutes a dynamic trading strategy, its performance is highly path-dependent.
Therefore, it is possible that the empirical results of those studies are more influenced by specific charcteristics of the underlying sample period rather than by the corresponding rebalancing algorithm under investigation. According to Brock, Lakonishok, and LeBaron (1992), this
danger of data snooping could be serious. For example, consider a market environment with a
very low volatility and a well-pronounced market trend. Perold and Sharpe (1988) show on a
theoretical basis that buy-and-hold must outperform any rebalancing strategy in this particular state of the market because the rebalancing algorithm requires buying past losers and investing the proceeds in past winners. More precisely, the regular reallocation to the worse
performing asset not only reduces the upside potential in prolonged upswing markets, but
also increases the downside potential in persistent cyclical downturns. This example illustrates that the resulting question of interest must be whether, on average, rebalancing leads to
a higher risk-adjusted performance in comparison to buy-and-hold.
By simulating 100,000 different 10-year return paths that could have been realized in the
past, our history-based simulation approach avoids this potential problem of data snooping.
We conduct a systematic analysis of the risk-adjusted performance of stock-bond portfolios
across different asset allocations and evaluate the impact of both rebalancing and buy-andhold by reporting statistical significance levels. In particular, we do not only apply the Sharpe
ratio, but also the Sortino ratio and the Omega measure in order to appropriately measure
risk-adjusted portfolio performance.
Sharpe Ratio Focusing on the financial market of the United States by way of example,
Panel A of Table 3 shows the average Sharpe ratios not only of all eleven different asset allocations, but also of all ten rebalancing strategies under investigation. As expected by the low
correlation between stocks and government bonds shown in Table 1, the diversification effect
is well-pronounced. For example, while buy-and-hold leads to an average Sharpe ratio of
10
0.67 for a 30/70 stock-bond portfolio, the average Sharpe of a 100/0 and a 0/100 stock-bond
portfolio is only 0.52 and 0.41, respectively. With regard to the underlying 30-year sample
period, an allocation of 30% stocks and 70% government bonds leads to the highest average
risk-adjusted portfolio performance for all ten rebalancing strategies. However, in results not
reported, the optimized asset allocation strongly depends on the period under investigation.
[Insert Table 3 here]
If the relative proportion of stocks accounts for at least 20%, buy-and-hold is outperformed in terms of Sharpe ratios by all nine rebalancing strategies. Panel B of Table 3 substantiates this observation by pointing out the average increase in risk-adjusted performance
of the respective strategy compared to buy-and-hold. As all three classes (periodic, threshold,
and range) and all three trading frequencies (yearly, quarterly, monthly) lead to a higher riskadjusted performance, on average, rebalancing seems to consistently provide a value-added
for institutional investors across different asset alloctions (if the stock allocation exceeds a
certain threshold). However, the question whether these differences are also statistically significant still remains unanswered.
Therefore, Panel A of Figure 1 shows the average Sharpe ratios of both quarterly periodic
rebalancing and buy-and-hold across different asset allocations for the financial market of the
United States. Panel B of Figure 1 presents the corresponding 10%-quantiles. If both boundaries are positive (negative), statistical significance is observed at the 10% level. As all parameters are linked by multiplication (four different classes of rebalancing, three different trading
intervals), we only show the risk-adjusted performance and the corresponding confidence
intervals for buy-and-hold as well as periodic quarterly rebalancing. Given an initial asset
allocation of 60% stocks and 40% government bonds, Dichtl et al. (2013) document that
quarterly periodic rebalancing provides the highest risk-adjusted performance on average.
With respect to the remaining rebalancing strategies classified in Table 2, the results differ
only slightly. In fact, comparing the risk-adjusted performance between rebalancing and buyand-hold, we observe a very similar pattern for all rebalancing strategies under investigation.
This observation also holds for the financial markets of the United Kingdom and Germany.
[Insert Figure 1 here]
11
As shown by Panel A of Figure 1, quarterly periodic rebalancing leads on average to higher Sharpe ratios compared to buy-and-hold if the stocks portfolio weight accounts for at least
30%. Panel B further indicates that this finding is statistically significant at the 10% level as
both interval boundaries are positive. In contrast, if the relative proportion of stocks falls below 20%, buy-and-hold outperforms quarterly periodic rebalancing. In results not reported,
this pattern disappears with both an increasing threshold and a decreasing trading frequency –
thus the more similar rebalancing becomes to buy-and-hold.
[Insert Figure 2 here]
We observe a similar pattern for the financial market of the United Kingdom as shown in
Panel A and Panel B of Figure 2. Again, quarterly periodic rebalancing produces significantly
higher Sharpe ratios compared to buy-and-hold if the portfolio weight of stocks accounts for
at least 30%. If the relative proportion of stocks falls below 30%, no statistical inference can
be drawn because the difference between rebalancing and buy-and-hold is lost in estimation
error. In contrast to the United States, the optimized asset allocation comprises of 20% stocks
and 80% government bonds. Moreover, adopting a buy-and-hold strategy, the average Sharpe
ratio of the optimal stock-bond portfolio with a value of 0.478 is substantially lower compared to the United States.
[Insert Figure 3 here]
Panel A of Figure 3 shows the average Sharpe ratios of both quarterly periodic rebalancing
and buy-and-hold across different asset allocations for the financial market of Germany,
while Panel B of Figure 3 depicts the corresponding 10%-quantiles. Again, our history-based
simulation results reconfirm the finding that rebalancing leads to a better risk-adjusted performance in terms of average Sharpe ratios across different asset allocations. We even observe statistical significance if the stock allocation exceeds 0%. Moreover, as hypothesized
by the discussion of the descriptive statistics in Table 1, the optimal stock-bond portfolio of
the German financial market consists of a higher proportion of government bonds in comparison to the optimal portfolio mix of the United States and the United Kingdom.
Sortino Ratio In contrast to the Sortino ratio, the Sharpe ratio considers all deviations
from the mean return – both negative and positive. However, as positive deviations constitute
an opportunity to generate an additional return on the invested capital, they are not expected
12
to be perceived as risk by investors. Defined by Sortino and Price (1994), the Sortino ratio
should better reflect investors’ risk perception by representing a risk-adjusted performance
measure that only takes negative deviations into account:
̅
,
(7)
where ̅ is the average return of strategy ,
∙ the corresponding probability density func-
tion, and the threshold return required by the investor. In order to distinguish between realized gains and losses, we set to zero. Illustrating the average Sortino ratio of quarterly periodic rebalancing and buy-and-hold for the United States, Panel A and Panel B of Figure 4
reconfirm our findings of the results stated above.
[Insert Figure 4 here]
Omega Measure Focusing on a two-asset class portfolio consisting of stocks and government bonds, portfolio returns are a linear function of stock returns if no transaction takes
place during the investment period. Therefore, buy-and-hold represents a linear investment
strategy. In contrast, rebalancing constitutes a concave investment strategy, as illustrated by
Perold and Sharpe (1988). On the one hand, the frequent reallocation to the weaker performing asset reduces the upside potential in upswing markets, thereby leading to an increase of
the portfolio return at a declining rate. On the other hand, a regular readjustment to the target
weights also reduces the downside protection in downswing markets, which causes a decline
of the portfolio return at an increasing rate. Ingersoll et al. (2007) show that investment strategies with a concave payoff lead to higher Sharpe ratios and Sortino ratios in comparison to
buy-and-hold by construction. Although this theoretical result does not contradict the finding
that rebalancing outperforms buy-and-hold, it points out that the possibility of portfolio performance manipulation must not be neglected, as stated by Ingersoll et al. (2007).
For this reason, we further apply the Omega measure as a robustness check. Developed by
Kaplan and Knowles (2004), the Omega measure represents the ratio of gains to losses relative to a predetermined threshold return:
(8)
Ω
1
,
13
where
denotes the return of strategy ,
∙ the cumulative density function, and the pre-
determined threshold return, which is again set to zero. In contrast to the Sharpe ratio and the
Sortino ratio, the Omega measure considers the entire return distribution of the underlying
investment strategy. As all moments are taken into account, any portfolio performance manipulation can be considered extremely difficult. Therefore, the use of the Omega measure
contributes to careful evaluation of risk-adjusted portfolio performance.
By way of example, Panel A of Figure 5 plots the average Omega measures of quarterly
periodic rebalancing and buy-and-hold against the underlying portfolio mix. Panel B of Figure 5 shows the corresponding 10%-quantiles. All in all, Figure 5 substantiates our prior findings that rebalancing significantly boasts risk-adjusted portfolio performance if stocks’ proportion exceeds a certain threshold. This proportion is subject to the country, the period under
investigation (results not reported), and the risk-adjusted performance measure. In the case of
Omega measures, it amounts to 20% for the United States and the United Kingdom and 0%
for Germany. In results not reported, we observe qualitatively similar results for the financial
markets of the United Kingdom and Germany.
[Insert Figure 5 here]
Overall, our analysis points out that cross-country differences have a substantial impact on
the level of risk-adjusted performance. Moreover, the period under investigation also affects
the level as well as the optimized asset allocation (results not reported). However, provided
that the stock allocation exceeds a certain threshold which is subject to the underlying country, our history-based simulation provides statistical evidence for all countries under investigation that rebalancing leads to a value-added for institutional investors.
6.
Conclusion
This study compares the impact on the risk-adjusted performance of stock-bond portfolios
across different asset allocations between rebalancing and buy-and-hold by reporting statistical significance levels. In order to simulate realistic market conditions and derive valuable
recommendations for investment practice, we apply the stationary bootstrap of Politis and
Romano (1994), which enables us to preserve most of the time series characteristics of financial time series (e.g., short-term momentum, fat tails, and empirical distribution). Modifying
the portfolio weight of stocks in steps of 10 percentage points from 0% to 100% stocks, our
14
analysis comprises eleven different stock-bond portfolio allocations for each of the three
countries under investigation.
If the portfolio weight of stocks exceeds 30%, our results provide statistical evidence that
rebalancing generates a superior risk-adjusted performance across all asset allocations for all
three countries and all three risk-adjusted performance measures under investigation. Despite
substantial cross-country differences, our analysis shows that rebalancing provides a valueadded for institutional investors of all three countries across different asset allocations. However, the optimized asset allocation strongly depends on both the country and the period under investigation.
15
REFERENCES
Adcock, Chris J., Nelson Areal, Manuel Armada, Maria Ceu Cortez, Benilde Oliveira, Florinda Silva. 2012. “Conditions under Which Portfolio Performance Measures Are Monotonic Functions of the Sharpe Ratio.” Working paper, University of Sheffield Management
School, Sheffield.
Arnott, Robert D., and Robert M. Lovell. 1993. “Rebalancing: Why? When? How Often?”
Journal of Investing, vol. 2, no. 1 (Spring): 5-10.
Black, Fischer, and Robert Litterman. 1992. “Global Portfolio Optimization.” Financial Analysts Journal, vol. 48, no. 5 (September/October): 28-43.
Brinson, Gary P., L. Randolph Hood, and Gilbert L. Beebower. 1986. “Determinants of Portfolio Performance.” Financial Analyst Journal, vol. 42, no. 4 (July/August): 39-44.
Brinson, Gary P., Brian D. Singer, and Gilbert L. Beebower. 1991. “Determinants of Portfolio Performance II: An Update.” Financial Analyst Journal, vol. 47, no. 3 (May/June): 4048.
Brock, William, Josef Lakonishok, and Blake LeBaron. 1992. “Simple Technical Trading
Rules and the Stochastic Properties of Stock Returns.” Journal of Finance, vol. 47, no. 5
(December): 1731–1764.
Cogneau, Philippe and Valeri Zakamouline. 2013. “Block Bootstrap Methods and the Choice
of Stocks for the Long Run.” Quantitative Finance, vol. 13, no. 9: 1443-1457.
DeMiguel, Victor, Lorenzo Garlappi, and Raman Uppal. 2009a. Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? Review of Financial Studies,
vol. 22, no. 5: 1915-1953.
DeMiguel, Victor, Lorenzo Garlappi, Francisco J. Nogales, and Raman Uppal. 2009b. “A
Generalized Approach to Portfolio Optimization: Improving Performance by Constraining
Portfolio Norms.” Management Science, vol. 55, no. 5 (May): 798-812.
Dichtl, Hubert, Wolfgang Drobetz, and Martin Wambach. 2013. “Testing Rebalancing Strategies for Stock-Bond Portfolios: Is the Value-Added of Rebalancing?” forthcoming in: Financial Markets and Portoflio Management.
Efron, Bradley, and Robert J. Tibshirani. 1998. “An Introduction to the Bootstrap.” New
York, Chapman & Hall/CRC.
Eling, Martin. 2008. “Does the Measure Matter in the Mutual Fund Industry?” Financial Analyst Journal, vol. 64, no. 3 (May/June): 54-66.
16
Fabozzi, Frank J., Petter N. Kolm, Dessislava A. Pachamanova, and Sergio M. Focardi. 2007.
“Robust Portfolio Optimization.” The Journal of Portfolio Management, vol. 33, no. 3
(Spring): 40-48.
Harjoto, Maretno A., and Frank J. Jones. 2006. “Rebalancing Strategies for Stocks and Bonds
Asset Allocation.” Journal of Wealth Management, vol. 9, no. 1 (Summer): 37-44.
Ibbotson, Roger G., and Paul D. Kaplan. 2000. “Does Asset Allocation Policy Explain 40, 90,
or 100 Percent of Performance?” Financial Analyst Journal, vol. 56, no. 1 (May/June): 2633.
Ingersoll, Jonathan, Matthew Spiegel, William Goetzmann, Ivo Welch. 2007. “Portfolio Performance Manipulation and Manipulation-Proof Performance Measures.” Review of Financial Studies, vol. 20, no. 5 (September): 1503-1546.
Jaconetti, Colleen M., Francis M. Kinniry, Yan Zilbering. 2010. Best Practices for Portfolio
Rebalancing. Vanguard, 1-17.
Kaplan, Paul D., and James A. Knowles. 2004. “Kappa: A Generalized Downside RiskAdjusted Performance Measure.” Journal of Performance Measurement, vol. 8, no. 3
(Spring): 42-54.
Kritzman, Mark. 2006. “Are Optimizers Error Maximizers?” The Journal of Portfolio Management, vol. 32, no. 4, (Summer): 66-69.
Ledoit, Oliver, and Michael Wolf. 2008. “Robust Performance Hypothesis Testing With the
Sharpe Ratio.” Journal of Empirical Finance, vol. 15, no. 5: 850–859.
Ledoit, Oliver, and Michael Wolf. 2011. “Robust Performance Hypothesis Testing With Variance.” Wilmott Magazine, vol. 55 (September): 86-89.
Leland, Hayne E., 1999. “Optimal Portfolio Management with Transactions Costs and Capital Gains Taxes.” Research Program in Finance, Working Paper RPF-290, University of
California, Berkeley.
Markowitz, Harry. 1952. “Portfolio Selection.” Journal of Finance, vol. 7, no. 1 (March): 7791.
Michaud, Richard O. 1989. “The Markowitz Optimization Enigma: Is ‘Optimized’ Optimal?”
Financial Analysts Journal, vol. 45, no. 1 (January/February): 31–42.
Norwegian Ministry of Finance. 2012. “The Management of the Government Pension Fund
in 2011.” Report No. 17 to the Storting.
17
Patton, Andrew, Dimitris N. Politis, and Halbert White. 2009. “Correction to “Automatic
Block-Length Selection for the Dependent Bootstrap” by Dimitris Politis and Halbert
White.” Econometric Reviews, vol. 28, no. 4: 372-375.
Perold, André F., and William F. Sharpe. 1988. “Dynamic Strategies for Asset Allocation.”
Financial Analysts Journal, vol. 44, no. 1 (January/February): 16-27.
Politis, Dimitris N., and Joseph P. Romano. 1994. “The Stationary Bootstrap.” Journal of the
American Statistical Association, vol. 89, no. 428 (December): 1303-1313.
Politis, Dimitris N., and Halbert White. 2004. “Automatic Block-Length Selection for the
Dependent Bootstrap.” Econometric Reviews, vol. 23, no. 1: 53-70.
Romano, Joseph P., and Michael Wolf. 2006. “Improved Nonparametric Confidence Intervals in Time Series Regressions.” Journal of Nonparametric Statistics, vol. 18, no. 2: 199214.
Sortino, Frank A., and Lee N. Price. 1994. “Performance Measurement in a Downside Risk
Framework.” Journal of Investing, vol. 3, no. 3 (Fall): 59-64.
Tokat, Yesim, and Nelson W. Wicas. 2007. “Portfolio Rebalancing in Theory and Practice.”
Journal of Investing, vol. 16, no. 2 (Summer): 52-59.
Tsai, Cindy S.-Y. 2001. “Rebalancing Diversified Portfolios of Various Risk Profiles.” Journal of Financial Planning, vol. 14 (October): 104-110.
Xiong, James X., Roger G. Ibbotson, Thomas M. Idzorek, and Peng Chen, 2010. “The Equal
Importance of Asset Allocation and Active Management” Financial Analysts Journal, , vol.
66, no. 2 (March/April): 22-30.
18
Table 1.
Descriptive Statistics
Asset
Statistics
United States
United Kingdom
Germany
Stocks
Mean (%)
Volatility (%)
Skewness
Kurtosis
Minimum (%)
Maximum (%)
10,45
15,77
-0,91
6,07
-23,85
12,47
10,84
16,14
-1,15
8,05
-30,02
13,72
8,75
22,06
-0,92
5,60
-28,67
19,02
Government Bonds
Mean (%)
Volatility (%)
Skewness
Kurtosis
Minimum (%)
Maximum (%)
8,57
7,91
0,05
3,66
-7,36
9,40
10,19
8,01
-0,06
4,45
-8,16
8,17
7,34
5,53
-0,29
3,26
-5,69
5,37
Cash (level)
Mean (%)
Volatility (%)
Skewness
Kurtosis
Minimum (%)
Maximum (%)
4,46
0,77
0,16
2,70
0,00
0,01
6,91
1,01
0,23
2,39
0,00
0,01
4,43
0,65
0,55
2,62
0,00
0,01
Correlations
Stocks/Bonds
Stocks/Cash
Bonds/Cash
0,039
0,081
0,076
0,19
0,06
0,11
-0,06
-0,02
0,07
Notes: This table presents the cross-sectional descriptive statistics of the stock, government bond, and money markets of the United States,
the United Kingdom, and Germany over the entire 30-year sample period from January 1982 to December 2011. Bonds denote government
bonds with a maturity of 10 years. Cash represents the corresponding 3-month money market rates. All statistics are calculated on a monthly
basis using continuous compounded returns. Mean, Volatility, Skewness, and Kurtosis denote the annualized mean return, volatility, skewness, and kurtosis. Skewness and Kurtosis are calculated as the third and fourth normalized centered moments. Minimum and Maximum are
the monthly minimum and maximum returns.
19
Table 2.
Classification of Implemented Rebalancing Strategies
Rebalancing Strategy
Frequency
Threshold
Reallocation
Classification
No.
Buy-and-Hold
No Adjustments
No Threshold
No Reallocation
Buy-and-Hold
1
Yearly Periodic Rebalancing
Quarterly Periodic Rebalancing
Monthly Periodic Rebalancing
Yearly
Quarterly
Monthly
No Threshold
No Threshold
No Threshold
Target Weights
Target Weights
Target Weights
Periodic
Periodic
Periodic
2
3
4
Yearly Threshold Rebalancing
Quarterly Threshold Rebalancing
Monthly Threshold Rebalancing
Yearly
Quarterly
Monthly
Threshold
Threshold
Threshold
Target Weights
Target Weights
Target Weights
Threshold
Threshold
Threshold
5
6
7
Yearly Range Rebalancing
Quarterly Range Rebalancing
Monthly Range Rebalancing
Yearly
Quarterly
Monthly
Threshold
Threshold
Threshold
Interval Boundaries
Interval Boundaries
Interval Boundaries
Range
Range
Range
8
9
10
Notes: This table presents all rebalancing strategies under investigation. The periodic rebalancing strategies 2, 3, and 4 are characterized by
a regular reallocation to the predetermined target weights at the end of each period. Strategies 5, 6, and 7 represent threshold rebalancing,
which is classified as periodic interval rebalancing with a strict adjustment to the target weights. In contrast, the range rebalancing strategies
8, 9, and 10 require a reallocation to the nearest edge of the predefined interval boundaries. A threshold of ±3% is applied to both threshold
rebalancing and range rebalancing.
Table 3.
Average Sharpe Ratios across Different Asset Allocations
Proportion of Stocks
Strategies
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Panel A: Average Sharpe Ratios
BAH
0.523
0.617
0.668
0.671
0.642
0.598
0.552
0.507
0.468
0.435
0.406
0.523
0.523
0.523
0.610
0.608
0.607
0.669
0.667
0.664
0.687
0.686
0.681
0.668
0.667
0.663
0.627
0.627
0.623
0.579
0.579
0.575
0.530
0.530
0.527
0.484
0.484
0.482
0.443
0.443
0.442
0.406
0.406
0.406
0.523
0.523
0.523
0.610
0.610
0.609
0.669
0.669
0.668
0.687
0.687
0.685
0.668
0.668
0.666
0.627
0.627
0.625
0.578
0.578
0.576
0.529
0.529
0.528
0.483
0.483
0.482
0.441
0.441
0.441
0.406
0.406
0.406
0.523
0.523
0.523
0.612
0.611
0.611
0.670
0.670
0.670
0.686
0.687
0.686
0.665
0.667
0.666
0.623
0.625
0.625
0.573
0.576
0.576
0.524
0.527
0.527
0.479
0.480
0.480
0.438
0.438
0.438
0.406
0.406
0.406
Periodic Rebalancing
Yearly
Quarterly
Monthly
Threshold Rebalancing
Yearly
Quarterly
Monthly
Range Rebalancing
Yearly
Quarterly
Monthly
Panel B: Average Increase in Sharpe Ratios in %
Periodic Rebalancing
Yearly
Quarterly
Monthly
0.0
0.0
0.0
-1.2
-1.4
-1.6
0.2
0.0
-0.5
2.4
2.2
1.6
4.0
3.9
3.2
4.8
4.8
4.0
4.9
5.0
4.2
4.4
4.5
3.8
3.4
3.4
3.0
1.9
1.9
1.7
0.0
0.0
0.0
0.0
0.0
0.0
-1.1
-1.2
-1.2
0.3
0.2
0.1
2.4
2.4
2.1
4.0
4.0
3.7
4.7
4.8
4.4
4.8
4.8
4.5
4.2
4.3
4.0
3.1
3.2
3.0
1.4
1.4
1.4
0.0
0.0
0.0
0.0
0.0
0.0
-0.7
-0.9
-1.0
0.4
0.4
0.3
2.2
2.4
2.3
3.5
3.8
3.8
4.1
4.5
4.4
4.0
4.4
4.4
3.3
3.8
3.8
2.2
2.5
2.6
0.7
0.8
0.8
0.0
0.0
0.0
Threshold Rebalancing
Yearly
Quarterly
Monthly
Range Rebalancing
Yearly
Quarterly
Monthly
Notes: Panel A of Table 3 presents the average Sharpe ratios of all rebalancing strategies under investigation across different asset allocations for the financial market of the United States. The underlying investment horizon is 10 years. Panel B of Table 3 shows the increase/decrease of the average Sharpe ratio of the corresponding rebalancing strategy in comparison to the average Sharpe ratio of a buyand-hold strategy. The sample period ranges from January 1982 to December 2011. Transaction costs are quoted at 15 bps per roundtrip.
1,000 simulations with an average block length of 2 are performed. Repeated simulations reveal that the results are stable.
20
Figure 1. Panel A.
Average Sharpe Ratios across Different Asset Allocations of the United States
0.70
0.65
Sharpe Ratio
0.60
0.55
0.50
0.45
0.40
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
90%
100%
Proportion of Stocks
BAH
Figure 1. Panel B.
Quarterly Periodic Rebalancing
10%-Quantiles of Average Sharpe Ratios of the United States
0.04
0.03
10%‐Quantiles
0.02
0.01
0.00
0%
10%
20%
30%
40%
50%
60%
‐0.01
‐0.02
Proportion of Stocks
Lower Boundary
Upper Boundary
70%
80%
21
Figure 2. Panel A.
Average Sharpe Ratios across Different Asset Allocations of the United Kingdom
0.50
0.45
Sharpe Ratio
0.40
0.35
0.30
0.25
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Proportion of Stocks
BAH
Figure 2. Panel B.
Quarterly Periodic Rebalancing
10%-Quantiles of Average Sharpe Ratios of the United Kingdom
0.04
0.03
10%‐Quantiles
0.02
0.01
0.00
0%
‐0.01
10%
20%
30%
40%
50%
60%
Proportion of Stocks
Lower Boundary
Upper Boundary
70%
80%
90%
100%
22
Figure 3. Panel A.
Average Sharpe Ratios across Different Asset Allocations of Germany
0.70
0.65
0.60
0.55
Sharpe Ratio
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
80%
90%
100%
Proportion of Stocks
BAH
Figure 3. Panel B.
Quarterly Periodic Rebalancing
10%-Quantiles of Average Sharpe Ratios of Germany
0.08
0.06
10%‐Quantiles
0.04
0.02
0.00
0%
10%
20%
‐0.02
30%
40%
50%
60%
Proportion of Stocks
Lower Boundary
Upper Boundary
70%
23
Figure 4. Panel A.
Average Sortino Ratios across Different Asset Allocations of the United States 2.60
2.40
2.20
Sortino Ratio
2.00
1.80
1.60
1.40
1.20
1.00
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
90%
100%
Proportion of Stocks
BAH
Figure 4. Panel B.
Quarterly Periodic Rebalancing
10%-Quantiles of Average Sortino Ratios of the United States
0.20
0.15
10%‐Quantiles
0.10
0.05
0.00
0%
‐0.05
10%
20%
30%
40%
50%
60%
Proportion of Stocks
Lower Boundary
Upper Boundary
70%
80%
24
Figure 5. Panel A.
Average Omega Measures across Different Asset Allocations of the United States
1.90
1.70
Omega Measure
1.50
1.30
1.10
0.90
0.70
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Proportion of Stocks
BAH
Figure 5. Panel B.
Quarterly Periodic Rebalancing
10%-Quantiles of Average Omega Measures of the United States
0.13
0.11
0.09
10%‐Quantiles
0.07
0.05
0.03
0.01
‐0.01
0%
10%
20%
30%
40%
50%
60%
‐0.03
‐0.05
Proportion of Stocks
Lower Boundary
Upper Boundary
70%
80%
90%
100%